Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 35, pp. 1-15.
Title: Some observations on the first eigenvalue of the p-Laplacian and
its connections with asymmetry
Author: Tilak Bhattacharya (Indian Statistical Inst. New Delhi, India)
Abstract:
In this work, we present a lower bound for the first eigenvalue
of the p-Laplacian on bounded domains in $\mathbb{R}^2$.
Let $\lambda_1$
be the first eigenvalue and $\lambda_1^*$ be the first eigenvalue
for the ball of the same volume. Then we show that
$\lambda_1\ge\lambda_1^*(1+C\alpha(\Omega)^{3})$, for some constant $C$,
where $\alpha$ is the asymmetry of the domain $\Omega$. This provides a
lower bound sharper than the bound in Faber-Krahn inequality.
Submitted September 3, 2000. Published May 16, 2001.
Math Subject Classifications: 35J60, 35P30.
Key Words: Asymmetry; De Giorgi perimeter; p-Laplacian;
first eigenvalue; Talenti's inequality.